The number of satellites increases with halo mass.
Thus, interaction is expected to be even more frequent
for halos of mass larger than \(10^{12}\, h^{-1}{\rm M_\odot }\) .
However, in these halos, most of the central galaxies are quiescent
galaxies, containing little cold gas,
so such interactions may not produce AGNs.
This, together with the fact the abundance of halos
decreases with halo mass, implies that only a small
fraction of optical AGNs reside in massive halos.
For halos with \(M_{\rm h}\ll 10^{12}\, h^{-1}{\rm M_\odot }\) , the mean number of
satellites is much smaller than unity, so that the
interaction probability is also small. This
suggests that the AGN fraction among low-mass
galaxies should also be lower, even though they
have plenty of cold gas and strong star formation.
Figure REF  shows the AGN fraction as a function
of halo mass, which is converted from AGN fraction as a function of
stellar mass based on the SHMR in [1]}.
We can see that AGN fraction reaches maximum
around \(10^{12}\, h^{-1}{\rm M_\odot }\)  and becomes
much lower at the low and high halo mass end, consistent with our analysis.
Note that the satellite number and \(t_{\rm it}\)  may vary
significantly among individual halos of a
given \(M_{\rm h}\) , so that the halos within which
significant interaction happens may have a broad
mass distribution. The number of satellites in a halo,
which increases with halo mass, the cold gas reservoir, and the
host halo abundance, which decreases with
halo mass, working together, may thus make halos
with \(\log {M_{\rm h, A}/\, h^{-1}{\rm M_\odot }}\sim 12\)  the
most favorable places for AGN activities.
